The lines y=(2/3)x+2 and y=(-2/3)x+12 are tangent to circle C at the points (3,4) and (3,10), respectively.
a) Find the coordinates of C and the radius of circle C. Explain your reasoning.
b)Write the standard equation of circle C.
Two radii of the circle are the lines perpendicular to the given lines. They are
y = -3/2 (x-3)+4
y = 3/2 (x-3)+10
They intersect at the center of the circle: (1,7)
The radius of the circle is r^2 = (1-3)^2 + (7-4)^2 = 13
The circle is (x-1)^2 + (y-7)^2 = 13
review the point-slope form of a line.
the line with slope m through (h,k) is
y-k = m(x-h)
Thanks for your response oobleck!
When finding radii equations, why did you say "(x-3)?"
To find the coordinates of the center and the radius of the circle, we need to analyze the given tangents to the circle.
Step 1: Find the slope of the tangent lines:
The lines y=(2/3)x+2 and y=(-2/3)x+12 are in the form y=mx+c, where m is the slope of the line. Comparing both equations, we see that their slopes are opposite reciprocals of each other. The slope of the tangent line perpendicular to the radius of the circle has a slope of -1/m.
For the line y=(2/3)x+2, the slope of the line perpendicular to it is -3/2.
For the line y=(-2/3)x+12, the slope of the line perpendicular to it is 3/2.
Step 2: Find the midpoint of the intersection of the two tangent lines:
Since the two tangent lines intersect at the point (3, 4), the midpoint of this point with the slope of -3/2 gives us the center of the circle, as the radius is perpendicular to the tangent line at the point of tangency.
Using the midpoint formula, the x-coordinate of the center is (3 + 3) / 2 = 3 and the y-coordinate of the center is (4 + 10) / 2 = 7.
Therefore, the center of the circle is C(3, 7).
Step 3: Find the radius of the circle:
To determine the radius, we need to find the distance from the center of the circle to one of the tangent points. Given that one of the tangent points is (3, 4), we can calculate the distance.
Using the distance formula, the radius of the circle is √((x2 - x1)² + (y2 - y1)²) = √((3 - 3)² + (4 - 7)²) = √(0 + 9) = √9 = 3.
Therefore, the radius of the circle is 3.
a) The coordinates of the center of the circle are (3, 7) and the radius of the circle is 3.
b) To write the standard equation of the circle, we use the formula (x - h)² + (y - k)² = r², where (h, k) are the coordinates of the center and r is the radius.
Substituting the values we found, the standard equation of the circle C is (x - 3)² + (y - 7)² = 3².